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Simplifying 2(3x + -2) + (x + -1)(x + -1) = 1 Reorder the terms: 2(-2 + 3x) + (x + -1)(x + -1) = 1 (-2 * 2 + 3x * 2) + (x + -1)(x + -1) = 1 (-4 + 6x) + (x + -1)(x + -1) = 1 Reorder the terms: -4 + 6x + (-1 + x)(x + -1) = 1 Reorder the terms: -4 + 6x + (-1 + x)(-1 + x) = 1 Multiply (-1 + x) * (-1 + x) -4 + 6x + (-1(-1 + x) + x(-1 + x)) = 1 -4 + 6x + ((-1 * -1 + x * -1) + x(-1 + x)) = 1 -4 + 6x + ((1 + -1x) + x(-1 + x)) = 1 -4 + 6x + (1 + -1x + (-1 * x + x * x)) = 1 -4 + 6x + (1 + -1x + (-1x + x2)) = 1 Combine like terms: -1x + -1x = -2x -4 + 6x + (1 + -2x + x2) = 1 Reorder the terms: -4 + 1 + 6x + -2x + x2 = 1 Combine like terms: -4 + 1 = -3 -3 + 6x + -2x + x2 = 1 Combine like terms: 6x + -2x = 4x -3 + 4x + x2 = 1 Solving -3 + 4x + x2 = 1 Solving for variable 'x'. Reorder the terms: -3 + -1 + 4x + x2 = 1 + -1 Combine like terms: -3 + -1 = -4 -4 + 4x + x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -4 + 4x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 4x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + 4x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 4x + x2 = 0 + 4 4x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 4x + x2 = 4 The x term is 4x. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4x + 4 + x2 = 4 + 4 Reorder the terms: 4 + 4x + x2 = 4 + 4 Combine like terms: 4 + 4 = 8 4 + 4x + x2 = 8 Factor a perfect square on the left side: (x + 2)(x + 2) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (x + 2) equal to 2.828427125 and -2.828427125.Subproblem 1
x + 2 = 2.828427125 Simplifying x + 2 = 2.828427125 Reorder the terms: 2 + x = 2.828427125 Solving 2 + x = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = 2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + x = 2.828427125 + -2 x = 2.828427125 + -2 Combine like terms: 2.828427125 + -2 = 0.828427125 x = 0.828427125 Simplifying x = 0.828427125Subproblem 2
x + 2 = -2.828427125 Simplifying x + 2 = -2.828427125 Reorder the terms: 2 + x = -2.828427125 Solving 2 + x = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = -2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + x = -2.828427125 + -2 x = -2.828427125 + -2 Combine like terms: -2.828427125 + -2 = -4.828427125 x = -4.828427125 Simplifying x = -4.828427125Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.828427125, -4.828427125}
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